Simulation 0

No association between longitudinal marker and mortality. Longitudinal marker has linear trend and independent error over time, no autoregressive component. Baseline mortality risk is time- and covariate-independent, i.e. constant over time and patients. Mortality is independent of the longitudinal trajectory.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
## 75604 24396
## .
##  0.5    1  1.5    2  2.5    3  3.5    4  4.5    5  5.5    6  6.5    7  7.5    8 
##  982  976  924  971  968  965  989  909  934  902  916  879  872  901  890  809 
##  8.5    9  9.5   10 10.5   11 11.5   12 12.5   13 13.5   14 
##  897  830  813  817  790  798  818  790  802  749  719  786

Patients at risk

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 1

No association between longitudinal marker and mortality. Longitudinal marker has linear trend and independent error over time, no autoregressive component. Baseline mortality risk is time-independent but does depend on covariates. Mortality is independent of the longitudinal trajectory.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
## 81148 18852
## .
##  0.5    1  1.5    2  2.5    3  3.5    4  4.5    5  5.5    6  6.5    7  7.5    8 
##  735  713  746  678  724  746  741  653  737  717  735  657  619  661  683  705 
##  8.5    9  9.5   10 10.5   11 11.5   12 12.5   13 13.5   14 
##  641  658  622  666  613  645  663  611  612  646  653  572

Patients at risk

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 2

The longitudinal marker trajectory is linear and has no autoregressive component. It is covariate dependent. The marker and mortality are positively associated based on a global dependence parameter, no local dependence changes over time. Mortality does not depend on covariates, i.e. the marker is the only component introducing association.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
##    53 99947
## .
##   0.5     1   1.5     2   2.5     3   3.5     4   4.5     5   5.5     6   6.5 
## 23262 17559 13768 10526  8126  6196  4750  3591  2868  2143  1648  1289   937 
##     7   7.5     8   8.5     9   9.5    10  10.5    11  11.5    12  12.5    13 
##   742   612   452   373   247   206   159   137    81    70    57    54    42 
##  13.5    14 
##    26    26

Patients at risk

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 3.1

The mortality risk and longitudinal marker trajectory have a quadratic trend. They are positively associated globally, without local changes.

The mortality risk is not covariate dependent.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
##   120 99880
## .
##   0.5     1   1.5     2   2.5     3   3.5     4   4.5     5   5.5     6   6.5 
## 16723 14002 11848  9875  8413  6965  5885  4850  4052  3326  2741  2224  1796 
##     7   7.5     8   8.5     9   9.5    10  10.5    11  11.5    12  12.5    13 
##  1433  1211   981   803   623   490   403   302   256   191   151   125    88 
##  13.5    14 
##    72    51

Patients at risk

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 3.2

The mortality risk and longitudinal marker trajectory have a quadratic trend. They are negatively associated globally, without local changes.

The mortality risk is not covariate dependent.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
##     5 99995
## .
##   0.5     1   1.5     2   2.5     3   3.5     4   4.5     5   5.5     6   6.5 
## 31204 21623 14887 10246  7038  4865  3263  2233  1562  1029   700   442   311 
##     7   7.5     8   8.5     9   9.5    10  10.5    11  11.5    13  13.5    14 
##   225   136    95    55    25    27    16     5     3     2     1     1     1

Patients at risk

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 4

The longitudinal marker and the mortality probability are positively associated and depend on the covariates. Both also have a quadratic trend in time.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
##   157 99843
## .
##   0.5     1   1.5     2   2.5     3   3.5     4   4.5     5   5.5     6   6.5 
## 17638 14857 11882  9256  8025  6606  5503  4679  4100  3397  2562  2279  1808 
##     7   7.5     8   8.5     9   9.5    10  10.5    11  11.5    12  12.5    13 
##  1577  1310   914   776   558   513   380   323   233   202   141   107    91 
##  13.5    14 
##    78    48

Patients at risk

### Variance across time for longitudinal marker The conditional variance within each partition, conditional on the lag-1 longitudinal marker value and the baseline covariates, remain constant. Considering the marginal variance across partitions, we expect a positive increase in variance as time progresses.

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 5.1

The longitudinal marker has no trend but a lag-1 autoregressive component. The baseline mortality is constant.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
## 48523 51477
## .
##  0.5    1  1.5    2  2.5    3  3.5    4  4.5    5  5.5    6  6.5    7  7.5    8 
## 1177 1088 1159 1097 1170 1008 1236 1051 1177 1110 1101  979 1089 1028 1069  998 
##  8.5    9  9.5   10 10.5   11 11.5   12 12.5   13 13.5   14 14.5   15 15.5   16 
## 1045  951  874 1074  858  917  860  929  916  889  830  810  814  758  831  757 
## 16.5   17 17.5   18 18.5   19 19.5   20 20.5   21 21.5   22 22.5   23 23.5   24 
##  750  747  788  783  825  865  800  758  728  694  779  767  812  698  642  670 
## 24.5   25 25.5   26 26.5   27 27.5   28 28.5   29 29.5   30 
##  698  703  609  699  698  615  571  574  671  637  563  683

Patients at risk

Variance across time for longitudinal marker

The longitudinal trajectory is stationary, as visible from the constant covariance.

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

Simulation 5.2

The longitudinal marker has no trend but a lag-1 autoregressive component. The baseline mortality is constant.

Baseline mortality risk

Longitudinal trajectory trend

Patients dying?

## .
## FALSE  TRUE 
## 49372 50628
## .
##  0.5    1  1.5    2  2.5    3  3.5    4  4.5    5  5.5    6  6.5    7  7.5    8 
## 1068 1156 1263 1176 1080 1099 1175 1049 1093 1033  966  990 1029  955  928  915 
##  8.5    9  9.5   10 10.5   11 11.5   12 12.5   13 13.5   14 14.5   15 15.5   16 
## 1025  934  992  890  841  899 1022  822 1012  916  948  801  828  808  827  838 
## 16.5   17 17.5   18 18.5   19 19.5   20 20.5   21 21.5   22 22.5   23 23.5   24 
##  782  878  698  807  765  774  844  718  670  701  704  689  764  753  692  630 
## 24.5   25 25.5   26 26.5   27 27.5   28 28.5   29 29.5   30 
##  660  624  599  654  669  580  606  656  612  648  552  521

Patients at risk

Variance across time for longitudinal marker

The longitudinal trajectory is not-stationary, as visible from the non-constant covariance over time.

Expected longitudinal marker trajectory for mortal and immortal cohort

Heterogeneity in longitudinal marker trajectory

## [1] 500